A note on Hadamard roots of rational functions

Suppose F is a polynomial and Σ_h≥0 F(b_h)X^h represents a rational function. If the b_h all belong to a field finitely generated over Q, then it is a generalization of a conjecture of Pisot that there is a sequence (c_h) with F(c_h) = F(b_h) for h = 0, 1, . . . so that also Σ_h≥0 c_hX^h represents a rational function. We explain the context of this Hadamard root conjecture and make some suggestions that might lead to its proof, emphasizing the apparent difficulties that have to be overcome and the ideas that might be employed to that end.